Continuing from my previous talk in February, I will present a second approach to constructing action-angle coordinates on multiplicity free Hamiltonian spaces.

The basic idea is to study gradient-Hamiltonian flows of toric degenerations of the base affine spaces G//N, where G is a complex reductive algebraic group. Toric degenerations of these spaces were constructed some time ago by Caldero. The problem of studying their gradient-Hamiltonian flows is a bit trickier than previous work on gradient-Hamiltonian flows because the fibers of the family are all non-compact and singular.

Once we have constructed these gradient-Hamiltonian flows, we can then combine them with the techniques of symplectic implosion and symplectic contraction to bootstrap the resulting integrable systems on G//N into integrable systems on multiplicity free Hamiltonian K-spaces, where K is the compact real form of G.

This is based on work in progress with Benjamin Hoffman.